Electronic theory for the normal state spin dynamics in Sr$_2$RuO$_4$: anisotropy due to spin-orbit coupling

Using a three-band Hubbard Hamiltonian we calculate within the random-phase-approximation the spin susceptibility, $\chi({\bf q},\omega)$, and NMR spin-lattice relaxation rate, 1/T$_1$, in the normal state of the triplet superconductor Sr$_2$RuO$_4$ and obtain quantitative agreement with experimental data. Most importantly, we find that due to spin-orbit coupling the out-of-plane component of the spin susceptibility $\chi^{zz}$ becomes at low temperatures two times larger than the in-plane one. As a consequence strong incommensurate antiferromagnetic fluctuations of the quasi-one-dimensional $xz$- and $yz$-bands point into the $z$-direction. Our results provide further evidence for the importance of spin fluctuations for triplet superconductivity in Sr$_2$RuO$_4$.